Math Snap

STEP 1

Assumptions1. We are given the integral $\int \cos (x)(13-\cos (x))^{8} \sin (x) dx$.
. We are asked to use a suitable change of variables (also known as substitution) to evaluate the integral.

STEP 2

The integrand has the form of a product of functions, one of which is a derivative of another. This suggests that we can use the substitution method to simplify the integral.
Let's choose $u =13 - \cos(x)$. This choice is motivated by the presence of $(13 - \cos(x))^8$ in the integrand.

STEP 3

Now, we need to find the derivative of $u$ with respect to $x$, which we'll denote as $du/dx$.
$$\frac{du}{dx} = -\sin(x)$$

STEP 4

We can rewrite the derivative $du/dx$ in terms of $dx$.
$$du = -\sin(x) dx$$

STEP 5

From the equation in4, we see that $\sin(x) dx = -du$. We can substitute this into the original integral.

STEP 6

Substitute $u$ and $du$ into the integral.
$$\int \cos (x)(13-\cos (x))^{8} \sin (x) dx = \int u^8 (-du)$$

STEP 7

The negative sign can be taken out of the integral.
$$\int u^ (-du) = -\int u^ du$$

STEP 8

Now, we can evaluate the integral of $u^8$ with respect to $u$.
$$-\int u^8 du = -\frac{1}{}u^ + C$$

Finally, we substitute back $u =13 - \cos(x)$ to get the answer in terms of $x$.
$$-\frac{}{9}u^9 + C = -\frac{}{9}(13 - \cos(x))^9 + C$$So, the solution to the integral is $-\frac{}{9}(13 - \cos(x))^9 + C$.